Construction of new solutions to field equations by using one nonseparable solution and one symmetry of the system

TL;DR

This paper introduces a method to generate numerous new solutions to various fundamental differential equations in physics by applying symmetry operators to a single nonseparable seed solution.

Contribution

It presents a novel approach that leverages symmetries to construct infinitely many solutions from one seed, applicable across multiple important physical equations.

Findings

Successfully generated new solutions for Schrödinger, diffusion, and paraxial equations.

Extended the method to Klein-Gordon, Dirac, Maxwell, and Einstein equations.

Demonstrated the approach's versatility with solutions in nonlinear general relativity.

Abstract

Symmetries of the field equations are used to construct infinitely many nontrivial linearly independent new solutions to different partial differential equations such as the Schroedinger, the diffusion, and the paraxial equations, among many others, including Klein Gordon, Dirac, Maxwell, Rarita Schwinger, linear Einstein field equations and even some especial seed solutions of fully nonlinear general relativity. The construction is done by applying one symmetry operator of the differential system to one nonseparable seed solution of the same system.